Equivariant stable homotopy theory. With contributions by J. E. McClure
type: Item
instance of: scholarly article
title: Equivariant stable homotopy theory. With contributions by J. E. McClure
zbMATH Open document ID: 0611.55001
published in: Lecture Notes in Mathematics
publication date: 1986-01-01T00:00:00Z
review text: The aim of this book is to lay the foundations of G-equivariant stable homotopy theory. Such theories have been in use in a more or less naive form for some time and with good success, but mostly they have been restricted to G-CW-complexes for finite groups G. The present construction of the theory uses May's approach to spectra and works for compact Lie groups G. We briefly indicate the basic notions. First one chooses a good category of spaces, namely the category of compactly generated weak Hausdorff spaces, which has the advantage of possessing also function spaces with the usual adjunction homeomorphisms besides the categorical limits and colimits. Now let G be a compact Lie group. A category of G-spectra is constructed for each ''G-universe'' U and ''indexing set'' A; here U is an orthogonal representation of G containing each finite-dimensional subrepresentation finitely often and containing a trivial representation, whereas A is a family of finite-dimensional subrepresentations of U which contains an ascending sequence exhausting U. A G-prespectrum D indexed on A then consists of based G-spaces D(V) for \(V\in A\) and based G-maps \(\sigma\) : \(\Sigma\) \({}^ wD(V)\to D(V\perp W)\) for V, \(V\perp W\in A\); it is called a G-spectrum if the adjoints \(\sigma\) : D(V)\(\to \Omega^ wD(V\perp W)\) are homeomorphisms. Maps are defined by the requirement of strict commutativity with the structure maps \(\sigma\). - The essential feature is now the existence of a left adjoint L to the inclusion functor from G-spectra to G-prespectra. Its importance is highlighted in Appendix A by a pointset-topology- construction in addition to the categorical existence proof; its usefulness comes from the possibility of transforming constructions of prespectra to constructions on spectra by applying L. The advantages (and disadvantages) of such an approach are as in the non- equivariant theory. (In the reviewer's opinion the advantages are here clearly in favour of this theory, e…
Mathematics Subject Classification ID: 55-02
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 57S10
Mathematics Subject Classification ID: 55N20
Mathematics Subject Classification ID: 18A40
zbMATH DE Number: 3988247
zbMATH Keywords: isotropic spectra for families of subgroups
zbMATH Keywords: G-equivariant stable homotopy theory
zbMATH Keywords: compact Lie groups
zbMATH Keywords: category of compactly generated weak Hausdorff spaces
zbMATH Keywords: category of G-spectra
zbMATH Keywords: G-prespectra
zbMATH Keywords: smash-products
zbMATH Keywords: function-spectra
zbMATH Keywords: change of groups
zbMATH Keywords: orbit- and fixed point constructions
zbMATH Keywords: G-CW-spectra
zbMATH Keywords: extended powers
zbMATH Keywords: operad ring spectra
zbMATH Keywords: equivariant duality transfer maps
zbMATH Keywords: Burnside ring
zbMATH Keywords: splitting theorems
zbMATH Keywords: Thom spectrum for a spherical fibration
zbMATH Keywords: equivariant Thom spectra of G-vectorbundles
author: L. G. jun. Lewis
author: J. P. May
author: Mark Steinberger
reviewed by: Erich Ossa
MaRDI profile type: Publication
DOI: 10.1007/BFB0075778
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description: scientific article; zbMATH DE number 3988247